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Intended for mathteacher
In the dark ages of modern computer software, about 49 years ago, I was applying what I called a magic formula for determining the amount of solar insolation that could be recovered on a flat surface at certain times of the day relative to local solar noon and various inclinations and orientations of said flat surface. The equation included various trig functions, including sin, cos, sinh and cosh.
The computer printout was providing valuable data until a certain point......................then BAM abort, abort abort.
The theory finally provided by the geniuses who ran the computing center was that under just the right circumstances the value for which a sin or cos was to applied had rounded off into a number in the computer that was a wee, tiny bit more than 1.00000. Thus, the computer did not know how to handle this. By intentionally modifying one of the inputs for that one case, the results for my entire request would be provided. TA DAH......MAGIC
I would be careful about saying what IBM engineers and their mathematician associates knew and didn't know about numbers. Most were PhD's with far more knowledge of the subtleties of math than I'll ever comprehend.
Consider this video:
Because the division algorithm says to choose a test quotient that, when multiplied by the divisor, gives a number less than the dividend, the test quotient that should be chosen is negative 8. The resulting remainder is 4.
It's not incorrect to modify the division algorithm and compute the quotient to be negative 7 and the remainder to be negative 1.
Both forms produce the same decimal quotient. In computer language development, it's a matter of convenience rather than a result of ignorance.
This source gives more insight: https://math.stackexchange.com/questions/2402838/why-do-so-many-computer-programming-language-implementations-have-trouble-with-t
My kids' middle school has 9 periods. English/Language Arts and Math are allocated 2 periods each. Science and Social Studies get 1 each. 1 period is a study hall with general music taking 2 days of the week. If you don't sign up to continue with music, I think it goes to straight study hall after 2 grading periods. PE/Health/Art share 1 period, switching between them every grading period.
Somehow I missed the news about the multitudinous options for the cube root of eight. Probably daydreaming during that class. I feel so incomplete now.
Dang, Melita. Just understanding the schedule would require serious critical thinking skills.
This is where the wondrous "i" comes into play. For those not familiar with i, the small letter i is used to denote a value equal to the square root of negative one. Thus, i squared is equal to negative one.
Now consider the fact that a number raised to a power such as 3,777 has 3.777 roots, as well. Chew on that for a while without looking like Daffy Duck after a stick of dynamite goes off in his mouth.
It gets worse. Ask ChatGPT this: How many solutions does x^11 = 3^7 have? You'll have to ask it 2 follow-up question:s: Are there no complex solutions? and What are the complex solutions?
It's interesting that one has to know what you said about the thousands of solutons to get a correct answer from ChatGPT and avoid the "that's what the computer said" mistake.
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Ask ChatGPT: What is the remainder when -36 is divided by 5?
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Two to the fourth power is sixteen, right? Thus sixteen also has four fourth roots. The obvious one being two. The semi-obvious solution is negative two. The other two fourth roots are complex numbers, so "i" is involved.
Hmmm. Doesn't that add up to 6 fourth roots?
I'm not sure if one plus one plus two equals four anymore. Once I start using imaginary numbers, my answers become imaginary as well.
What's more fun is that they're twins and I specifically asked the school to give them different schedules. Beyond the 'special' classes (gym, art, music, health, etc.), they share one teacher (still different periods). One kid's doing great and the other is doing okay. They're not best friends, so I thought it better to give them some separation at school.
It was really exciting signing up for teacher conferences. I think I ended up with 12?
